- predefined_virtuals: an addition

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / substitution / tps_lift.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma
index e453c4099cd4a8b06f7f1a63f507672fba04e989..6f712e8a72aad541cb4d5e56e35d37731960253b 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma
@@ -12,39 +12,48 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic_2/substitution/ldrop_ldrop.ma".
-include "Basic_2/substitution/tps.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/tps.ma".
 
 (* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
 (* Advanced inversion lemmas ************************************************)
 
-fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → e = 1 →
-                           ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 → 
+                     ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
+#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
 [ //
-| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
-  >(le_to_le_to_eq … Hdi ?) /2 width=1/ -d #K #V #HLK
-  lapply (ldrop_mono … HLK0 … HLK) #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
-  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 // /2 width=1/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
-  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
+| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
+  elim (lt_or_ge i (d+1)) #HiSd
+  [ -Hide1 -HV0
+    lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
+    lapply (ldrop_mono … HLK0 … HLK) #H destruct
+  | -V -Hdi /2 width=4/
+  ]
+| /4 width=3/
+| /3 width=3/
 ]
 qed.
 
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ▶ T2 →
+lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
+                  ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
+/2 width=3/ qed-.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
                         ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-/2 width=8/ qed-.
+#L #T1 #T2 #d #HT12 #K #V #HLK
+lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
+lapply (tps_inv_refl_O2 … HT12) -HT12 //
+qed-.
 
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
                    ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
                    ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    dt + et ≤ d →
-                   L ⊢ U1 [dt, et] ▶ U2.
+                   L ⊢ U1 ▶ [dt, et] U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -65,11 +74,11 @@ lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
 ]
 qed.
 
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
                    ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
                    ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    dt ≤ d → d ≤ dt + et →
-                   L ⊢ U1 [dt, et + e] ▶ U2.
+                   L ⊢ U1 ▶ [dt, et + e] U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -98,11 +107,11 @@ lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
 qed.
 
 (* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
                    ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
                    ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    d ≤ dt →
-                   L ⊢ U1 [dt + e, et] ▶ U2.
+                   L ⊢ U1 ▶ [dt + e, et] U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -122,10 +131,10 @@ lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
 qed.
 
 (* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         dt + et ≤ d →
-                        ∃∃T2. K ⊢ T1 [dt, et] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+                        ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -149,10 +158,10 @@ lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
 ]
 qed.
 
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         dt ≤ d → d + e ≤ dt + et →
-                        ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+                        ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -187,10 +196,10 @@ lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
 qed.
 
 (* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         d + e ≤ dt →
-                        ∃∃T2. K ⊢ T1 [dt - e, et] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+                        ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -221,7 +230,7 @@ qed.
 
 (* Basic_1: was: subst1_gen_lift_eq *)
 lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
-                        L ⊢ U1 [d, e] ▶ U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+                        L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
 #L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
 [ //
 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
@@ -253,10 +262,10 @@ qed.
                                         (le d i) -> (lt i (plus d h)) ->
                                        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
 *)
-lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
-                        ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                        d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
-                        ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+                           ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                           d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+                           ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
 elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
 lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
@@ -264,11 +273,22 @@ lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
 elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
 qed.
 
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
                            ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                            dt ≤ d → dt + et ≤ d + e →
-                           ∃∃T2. K ⊢ T1 [dt, d - dt] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+                           ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
 lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
 elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
 qed.
+
+lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+                           ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                           dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+                           ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
+elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
+lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
+lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
+qed.